Modeling Tides with Trigonometry
Date: 03/07/2001 at 21:53:58 From: Saundra Holseth Subject: Trigonometry, sinusoids This is a story problem, with multiple parts. I think that if you can help me get started, I can finish. The depth of water at the end of a pier varies with the tides throughout the day. Today, January 23, the high tide occurs at 4:15 A.M. with a depth of 5.2 meters. The low tide occurs at 10:27 A.M. with a depth of 2.0 meters. A. Find a trigonometric equation that models the depth of the water t hours after midnight, and graph it. B. Find the depth of the water at noon. C. A large boat needs at least 3 meters of water to moor at the end of the pier. During what time period after noon can it safely moor? Show this point on a graph in red. D. What is the first time after midnight on February 1 when high tide occurs? Show this point on your graph in blue. (Hint: set your time scale on the x-axis to run from midnight on Jan. 23 to noon on Feb. 1.)
Date: 03/08/2001 at 09:11:10 From: Doctor Wolfson Subject: Re: Trigonometry, sinusoids Hi Saundra, Let's start by writing down exactly what variables we can work with to find the answer. The general form of the trig function is: y = A cos(Bx + C) + D I'll explain later why I used cosine instead of sine. Here, A is the amplitude from the middle line to the peak, B is the frequency divided by 2pi radians, or 360 degrees, C is the horizontal (phase) offset, and D is the vertical offset. Now let's find each of these. A is half the difference between the high tide value, and the low tide value, and those numbers are in the problem. B is related to the frequency. The amount of time it takes to go from high tide to low tide is half of one period, so if you subtract those times, and double the answer, you'll find the full period. To find the value of B, start with 2pi radians (or 360 degrees) and divide by that full period; and you'll have B. C tells us when the graph "starts." I decided to use cosine because if C = 0, then it hits "high tide" at x = 0, and we are given the times for high tide (instead of "middle tide"). Now you can find what fraction of a period after midnight high tide occurs, multiply by 2pi (or 360), and write C. Remember that since we want to shift the graph to the right, it will have a minus sign. D is how far above 0 the middle line is. This is just the average of high and low tides. So now you have your trig equation. You can find a noon tide height by just evaluating the equation at the right time. For the next part, find where the graph crosses y = 3 (or subtract 3 from the function and find the zeros.) For the last part, you know that high tide will occur after an integer number of periods from the beginning high tide, so just find out how many periods it takes to get to Feb. 1. I hope this helps - feel free to write back if you'd like further clarification. - Doctor Wolfson, The Math Forum http://mathforum.org/dr.math/
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